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Acknowledgement
This research was supported by the Croatian Science Foundation Project No. IP-2020-02-9614 Search for Quantum spacetime in Black Hole QNM spectrum and Gamma Ray Bursts. The work of N.K. is supported by Project 451-03- 136/2025-03/200162 of the Serbian Ministry of Science, Technological Development and Innovation.
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Noncommutative (NC) geometry provides a novel approach to probe quantum gravity effects in black hole spacetimes. This work explores Dirac quasinormal modes (QNMs) of a deformed Reissner-Nordström ¨om black hole, where noncommutativity induces an effective metric with an additional (r−φ) component. Employing a semiclassical model equivalent to an NC gauge theory, we investigate the dynamics of massless Dirac fields and calculate their QNM frequencies using the continued fraction method, enhanced by Gauss elimination to address the six-term recurrence relations. Our results demonstrate notable shifts in oscillation frequencies and damping rates relative to the commutative Reissner-Nordström ¨om case, exhibiting a distinctive Zeeman-like splitting in the QNM spectrum driven by the NC parameter.
Black holes, once confined to the realm of mathematical curiosity, have emerged as empirical laboratories for exploring gravity in its most extreme form. The detection of gravitational waves by LIGO, Virgo, and KAGRA has transformed these compact objects into astrophysical tools, opening a new observational frontier often referred to as gravitational-wave astronomy [1]. Within this framework, the quasinormal modes (QNMs) of black holes—those damped oscillations that arise from perturbations—have become central observables. These modes, akin to the resonant frequencies of a bell, carry detailed information about the geometry of the black hole and the underlying gravitational theory. They offer a novel means to probe not only classical general relativity but also its possible extensions at the quantum level. Indeed, in the emerging field of gravitational spectroscopy, one of the key motivations is to investigate whether QNMs can resolve fine structures in the spacetime spectrum—structures that may reveal imprints of quantum gravity. Among the diverse proposals for quantum gravity, noncommutative (NC) geometry provides a particularly elegant and mathematically rich framework. Inspired by developments in quantum field theory and string theory, NC geometry replaces the classical notion of spacetime with a NC algebra of functions, thereby introducing a minimal length scale [15]. This “fuzziness” at the Planck scale leads to modifications in the local structure of spacetime that could, in principle, manifest in strong gravity regimes. In the semiclassical approach adopted here, one considers deformations of the spacetime metric induced by NC gauge field interactions while treating the gravitational background classically. A particularly tractable case is obtained when the NC deformation is encoded
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In this work, we investigated the propagation of a Dirac field in a NCdeformed RN black hole background and computed the corresponding QNM frequencies using Leaver’s continued fraction method. Our analysis was perturbative in the NC parameter a and is valid to linear order in a. The results highlight that NC geometry introduces distinct signatures in the QNM spectrum, most notably a Zeeman-like splitting of frequencies. Such effects suggest that QNMs can serve as potential probes of spacetime noncommutativity, enriching the framework of gravitational spectroscopy. Our study focused on massless fermions and non-extremal black holes. Future investigations should consider extensions to massive Dirac fields and extremal black hole limits, which require separate analytical treatment due to the breakdown of the current method in those regimes
Conclusion
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